A fractal–fractional differential model for distributed denial-of-service attack dynamics

Abstract

Distributed denial-of-service (DDoS) attacks have become a major threat to the stability of critical infrastructure networks, where even short service disruptions can lead to severe operational and economic consequences. To better capture the complex dynamics of these attacks, we extend an existing epidemic-based DDoS model by employing the fractal–fractional (FF) Atangana–Baleanu (AB) operator, which effectively accounts for memory effects, network heterogeneity, and irregular traffic patterns commonly observed in cyber environments. Within this framework, we establish the existence and uniqueness of solutions and examine the Ulam– Hyers stability of the proposed system. The local stability of both infection-free and endemic equilibria is assessed to identify the conditions under which the network can maintain normal operation. Numerical simulations are performed using the Adams–Bashforth method for various combinations of fractional and fractal orders. The results show that the FFAB formulation captures slower decay, extended memory, and more realistic transient dynamics than its classical counterpart. These findings demonstrate that incorporating FF dynamics offers a more flexible and accurate representation of DDoS propagation and quarantine- based mitigation, providing valuable insights for enhancing the resilience of modern cyber-infrastructure systems.

Keywords

Atangana–Baleanu operator / Cybersecurity modeling / Distributed denial-of-service / Existence and uniqueness / Fractal–fractional derivative

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